STRAIGHT LINE
► Definition: Straight line is a curve such that every point on the line segment joining any two points in it, entirely lies on it.
🌑 General Equation of Straight Line :
Any equation of first degree of the form ax+by+c=0, where a ,b ,c are constants always represents a straight line (at least one out of a, b is non-zero)
🌑 Slope/Gradient of a Line :
If a line makes an angle θ with the positive direction if x-axis then tanθ is said to be the slope or gradient of the line. We generally denote by ‘m’, where the value of θ is between 0° to 180° degree.(θ not equals to 90° degree).
⬜ The slope of a line parallel to x-axis is m= tan0 =0°
⬜ The slope of a line parallel to y-axis is not defined
⬜ If a line is equally with the axis then its slope is 1 or -1
⬜ If A(X₁,Y₁) and B(X₂,Y₂) be any two points, then the slope of AB is `(Y_2-Y_1)/(X_2-X_1)=\tan\theta=dy/dx`
⬜ Slope of a line ax+by+c=0, b not equals to 0, is `-(a/b)=-[(coefficient \ of \ x)/(coefficient \ of \ y)]` .
Note: If a straight line passes through the point (X₁,Y₁) and it's gradient is ‘m’, then the equation of the straight line is Y-Y₁ = m (X-X₁) .
🌑 Different forms of Straight Line :
⬜ General Form :
ax+by=c
● In this equation for any values of a, b, c we get a point on the straight line.
● C=0, the straight line passes through origin.
⬜ Gradient-Intercept Form:
● If a line L has a slope ‘m’ and its intercept on Y-Axis is C, then the equation of line L is given by y=mx+c
● The distance where the straight line cuts the x-axis from the origin is called ‛X Intercept’ and the distance from origin where the straight line cut the y-axis is called ‘Y Intercept.’
● It is not a general form cause it cannot define the straight line parallel to Y axis.
⬜ Two-Point Form :
● The equation of a straight line passing through two given points (X₁,Y₁) and (X₂,Y₂) is given by`(Y-Y_1)=[(Y_2-Y_1)(X-X_1)]/(X_2-X_1)`
⬜ Intercept Form :
● The equation of the line which cuts off intercepts d and c on x-axis and y-axis respectively is given by `(x/d)+(y/c)=1`
● Problems
■ We can not calculate the straight lines
1. Which passes through origin
2. Which is parallel of X-axis and
3. Which is parallel to Y-axis
Examples:
3x+4y=12 general Form
`y=-(3/4)x+3` gradient intercept form
`(x/4)+(y/3)=1` Intercept Form.
⬜ Normal Form :
● The equation of a straight line upon which the length of perpendicular from the origin is ‘p’ and the perpendicular makes an angle α with the positive direction of X-axis is given by x cosα + y sinα = p
● In the normal form of equation of a straight line ‘p’ is always taken as positive and α is measured from positive direction of x-axis in anti-clock wise direction between 0 and 2Ï€.
⬜ Symmetric Form :
● `(x-x_1)/\cos\theta= (y-y_1)/\sin\theta` [ the value of θ lies between 0 to Ï€] [θ: Angle made by the straight line with the positive direction of X-axis, Fixed point (x₁,y₁) on the straight line]
⬜ Parametric Form :
● The equation of line passing through a given point P(X₁,Y₁) and making an angle θ with positive direction of X-axis is given by `(X-X_1)/\cos\theta= (Y-Y_1)/\sin\theta=r` [where ‘r’ is the distance between the variable point Q(X,Y) and the fixed point P(X₁,Y₁)]
● Any point on the straight line will be of the form (X₁+rcosθ, Y₁+rsinθ) for different values of ‘r’, we will get different points on the line.
● If P(X,Y) is any point on the line which makes an angle θ with the positive direction of X-axis, then there will be two points on the line at a distance 'r' from P(X₁,Y₁). One will be relatively upward if ‘r’ is taken positive and other will be relatively downward if ‘r’ is taken negative.
Note:
1. Equation of a line parallel to X-axis is y=±a(a ∈ R)
2. Equation of a line parallel to Y-Axis is X=±b(b ∈ R)
🌑 Important Formula and Facts:
⬜ Distance Formula:
Distance between two points X(a,b) and Y(c,d)
D(X,Y)=`[(c-a)^2+(d-b)^2]^(1/2)`
⬜Section Formula:
If a point P devides the line segment joining two points. X(a,b) and Y(c,d)in the ratio m:n internally or externally it is given by this formula
● Internally P= `((mc+na)/(m+n),(md+nb)/(m+n))`
● Externally P= `((mc-na)/(m-n), (md-nb)/(m-n))`
⬜ Centroid:
centroid of a triangle joining (a,b), (c,d), (e,f) is given by `((a+c+e)/3, (b+d+f)/3)`
⬜ Area of a triangle:
Area of triangle joining (a,b), (c,d), (e,f) is given by absolute value of
⬜ Equation of line parallel and perpendicular to a given line:
● Let, ax+by+c=0 be a given line then the equation of parallel line to the given line is ax+by+d=0; where, d not equals to C, it is any arbitrary constant.
● The equation of the line perpendicular to ax+by+c=0 is given by bx-ay+d=0; where d not equals to C is any arbitrary constant.
⬜ General Equation of family of lines through the intersection of two given lines:
● Let the two lines are L₁: ax+by+c=0 and L₂: dx+ey+f=0, then the equation of lines passing through the intersection of L₁ and L₂ is given by
L₁+ UL₂=0 or, L₂+ UL₁=0
(ax+by+c)+U(dx+ey+f)=O
or, (dx+ey+f)+U(ax+by+cz)=0;
• Where U is a parameter.
⬜ Angle between two straight line :
● Let θ be the angle between the lines L₁=0 and L₂=0, where L₁: y=m₁x+ c₁, L₂: y=m₂x+c₂, then tanθ=`|(m_1- m_2)/(1+m_1m_2)|`
● If m₁, m₂ are slopes of two straight lines then
• Parallel lines, m₁=m₂
• Perpendicular lines, m₁m₂= -1
⬜ Change of Axis:
● Translation of Axis:
If we shift the origin to (a, b) without rotation of x, y-axis then co-ordinate of a point (x, y), it in old system will change to (x-a, y-b) in new system.
● Rotation of Axis:
If we rotate the axis through an angle θ in the anti-clockwise direction without changing the origin, then co-ordinate of (x, y) in old system will change to (xcosθ + ysinθ, ycosθ – xsinθ) in new system.
⬜ Angle Bisectors:
● Two angle bisectors of two intersecting straight line ax + by + c = 0 and dx + ey + f = 0 `(a/d≠ b/e)` are given by `(ax+by+c)/\sqrt{a^2+b^2}=(dx+ey+f)/\sqrt{d^2+e^2}`
⬜ Relative position of a point:
● The two points (X₁, Y₁) and (X₂, Y₂) lie on the same or opposite side of a straight line ax + by + c = 0 according to aX₁ + bY₁ + C₁ and aX₂ + bY₂ + C₂ have the same sign or opposite signs respectively.
⬜ Distance from a point to a straight line:
● The distance of point (X₁, Y₁) to a straight line ax + by + c = 0 means the perpendicular distance of the point to the straight line is given by `|(aX_1 + bY_1 + c)/(a^2 + b^2)^(1/2)|`
⬜ Distance between two parallel straight line:
● The distance between ax + by = C₁ and ax + by = C₂ means the perpendicular distance between the two is given by `α/(a^2 + b^2)^(1/2)`
• α = |C₁ - C₂| if they are on the same side of the origin
• α = |C₁| + |C₂|, if origin lies between them.
⬜ Concurrency of Lines:
● Three or more lines are said to be concurrent if they meet/ intersect at the same point. Let the three lines be
• L₁: a₁x + b₁y + c₁ = 0,
L₂: a₂x + b₂y + c₂ = 0,
L₃: a₃x + b₃y + c₃ = 0
• These lines will be concurrent if
● The condition for the lines L = 0, M = 0, N = 0 to be concurrent is that three constants a, b, c (not all zero at the same time) can be obtained such that aL + bM + cN = 0
⬜ Mirror Image of a point with respect to a given line:
● Let L: ax + by + c = 0 be the equation of line. Let (x₂, y₂) be the mirror image of (x₁, y₁) then `(x_2 - x_1)/a = (y_2 - y_1)/b = -2((ax_1 + by_1 + c)/(a^2 + b^2)) `
● Note: If (x₂, y₂) be the foot of the perpendicular from (x₁, y₁) to the line ax + by + c = 0 then, `(x_2 - x_1)/a = (y_2 - y_1)/b = - ((ax_1 + by_1 + c)/(a^2 + b^2))`
⬜ In case of bisectors if two given straight line a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 without loss of generality we can assume c₁,c₂ > 0 then with the equation of bisectors as
● `((a_1x + b_1y + c_1)/(a_1^2 + b_1^2)^(1/2)) = \pm ((a_2x + b_2y + c_2)/(a_2^2 + b_2^2)^(1/2)) ` we can follow the following trick(s) as
• If a₁a₂ + b₁b₂ < 0, + sign will give the bisector of acute angle and – sign for obtuse.
• If a₁a₂ + b₁b₂ > 0, - sign will give the bisector of acute angle, + sign for obtuse.
• + Sign will always give the bisector of that angle where origin lies
• If (a₁x + b₁y + c₁) (a₂x + b₂y + c₂) > 0 then + sign will give the bisector of that angle (x, y) lies.
⬜ Pair of Straight Lines :
● (a) For the general equation of two variable (x, y) of degree two: ax² + 2hxy + by² + 2gx + 2fy + c = 0 to represent a pair of straight line if the following condition holds.
● (b) Pair of straight line through the origin is represents by
ax² + 2hxy + by² = 0 with h² > ab
⇒ (y - m₁x)(y - m₂x) =0
⇒ `m_1 + m_2 = (2h/b), m_1m_2 = a/b` [If at least one of a not equals to 0, b not equals to 0 ] Angle between the pair of straight line is given by
⬜ Area of a polygon with vertices A₁ (X₁, Y₁); A₂ (X₂, Y₂) ; A₃(X₃,Y₃);…… Aâ‚™ (Xâ‚™, Yâ‚™) is equals to
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